In this Chapter, the MISO interference channel with local CSIT at each transmitter was considered. Maximizing a virtual SINR was proposed as a beamforming strategy which uses only the local information and avoids iterative algorithms which require information exchange between different transmitters. The validity of such a scheme was discussed by relating it to existing literature on the subject and its Pareto optimality for a two link case was shown. In general, if the objective is sum rate maximization, this scheme performs quite well if there are at least as many antennas at each
4.7 Conclusion 91
−20 −15 −10 −5 0 5 10 15 20
0 2 4 6 8 10 12 14 16 18 20
SNR (dB)
Sum rate (bits/sec/Hz)
MISO ADP, random initial beamforming MISO ADP, VSINR initial beamforming VSINR beamforming
(a)K= 2
−30 −20 −10 0 10 20 30 40
0 5 10 15 20 25
SNR (dB)
Sum rate (bits/sec/Hz)
MISO ADP, random initial beamforming MISO ADP, VSINR initial beamforming VSINR beamforming
(b) K= 3
Figure 4.5: Average sum rate vs. SNR for i.i.d. Rayleigh channels and Nt= 2.
92 Chapter 4 Coordinated MISO IC using the VSINR Framework transmitter as there are active links. When this is no longer the case, such a scheme, while still performing better than MRT or interference minimiz-ing transmit strategies, performs much worse than the optimal scheme. In such a case the sum rate is maximized by having a fraction of the users transmit at less than full power. In fact at high SNR, some links would simply turn off. The degrees of freedom afforded by a scheme that relies on beamforming alone is in fact min (Nt, K) so that if Nt < K, different interference avoidance schemes may be needed to be able to serve more con-current users: interference alignment in the time or frequency domain (for a frequency selective channel) [52], or by restricting signals to be real in-stead of complex-valued [53] for example would be needed to achieve higher degrees of freedom [46]. Attaining these degrees of freedom does not seem to be possible with local CSI alone [52], not without resorting to iterative schemes or perhaps by devising some distributed power control scheme.
4.A 93
4.A Proof of Theorem 3
To simplify expressions, in what follows ¯k is used to denote the ’other’
user/base station index (i.e., ¯k= mod (k,2) + 1, for k∈ {1,2}).
We show that the rate region achieved by the parametrization given in Cor-rolary 1 of the two-link Pareto boundary can also be achieved by varying the α’s in their feasible region (R+) and maximizing the corresponding virtual SINRs.
Maximizing the virtual SINR of (4.7):
uk = arg max
kuk2=1
|hk,ku|2
1/ρ+αkk¯|h¯k,ku|2. (4.19) Proposition 3. The solution of problem (4.19) can be written as:
uk=p h¯k,k and its the null space, respectively:
Πkk¯ = hH¯k,kh¯k,k
kh¯k,kk2 , Π⊥¯kk=INt −Πkk¯ . (4.21) Proof. Similar to that of Proposition 1 in [48].
Define:
ak =kΠkk¯ hHk,kk2 bk =kΠ⊥kk¯ hHk,kk2
ck =ρkh¯k,kk2 (4.22)
Proposition 4. ζk that solves (4.19)is given by:
ζk= ak
94 Chapter 4 Coordinated MISO IC using the VSINR Framework
Similarly,
|h¯k,kuk|2=ζkkh¯k,kk2 (4.25) Thus the virtual SINR is equal to:
γkvirtual = ρ√
akζk+p
bk(1−ζk)2
1 +αkk¯ζkck (4.26) One can easily verify that this ratio is maximized for the value specified in (4.23).
Proposition 5. In terms of the NE and ZF beamforming vectors, (4.20) can be rewritten as:
uk= λkuN Ek + (1−λk)uZFk We need to show that this is in fact of the form given in (4.27), i.e. that the following equalities hold for someλk:
λ2k
where we replaced the denominator of (4.27) by its value in terms of the parameters defined in (4.22).
One can verify that λk as given by (4.28) above satisfies both these equations.
4.B 95 Combining propositions 4 and 5, we complete the proof. Plugging (4.23) into (4.28), we get:
λk= 1
αkk¯ckq
bk
ak+bk + 1 (4.30)
Clearly this is a decreasing function of αk¯k. It is easy to check that for αk¯k= 0,λk= 1 and that asαk¯k→ ∞,λk→0.
4.B Minimizing total transmit power subject to SINR and per transmitter power constraints
Consider the problem of minimizing total transmit power such thatγkt,k= 1, . . . , K are achieved at each of the users, while meeting the individual power constraints at each transmitter.
minimizeα
This can easily be shown to be equivalent to a convex optimization problem (see [51, 54] for example).
The corresponding Lagrangian is equal to
α where λk ≥0 denotes the Lagrange coefficient associated with power con-straintk, andµk≥0 denotes the Lagrange coefficient associated with SINR constraint k.
The dual problem is that of maximizing σ2
K
X
k=1
µk (4.33)
96 Chapter 4 Coordinated MISO IC using the VSINR Framework
being positive semi-definite (PSD), andPK
k=1(1−λk)P = 0. In other words,
For the above constraint to hold for any wˆk, it must hold for the one that maximizes its right hand side. Moreover, one could show that at the optimum of the dual, the constraints (4.36) must be met with equality. Thus optimalˆuk could be the MMSE filters given by
Going back to the primal problem, from the KKT conditions we have that
Thus, any solution for the above power minimizaton problem subject to SINR constraints is up to a scalar the solution of an equivalent uplink SINR maximization problem as expressed by the left-hand side of (4.36). This
4.C 97 includes the SINR values that lie on the Pareto boundary. Since we know that in this case the optimal uk have norm 1 and therefore the optimalwk will have normP, these will indeed be the solutions of maximizing a virtual SINR as given by (4.7) such that
λk ck = 1
ρ and µj
ck =αkj, j6=k (4.40) for some ck > 0. Moreover, since the duality gap is zero and the optimal α= 1,
Kρ=
K
X
k=1
µk. (4.41)
4.C Proof of Theorem 4 for two links
For any u1,u2 of the form (4.20), one can show that SINRs are given by:
γk=ρ(√
akζk+p
bk(1−ζk))2 1 +ζ¯ic¯i
(4.42)
A rate pair is Pareto optimal if one cannot increase one of the rates without necessarily decreasing the other. Note that any point on Pareto boundary has to have the corresponding (ζ1, ζ2) pair in the region defined byζk∈h
0,aak
k+bk
i, k= 1,2: this is so since for higherζkit is always possible to achieve higher useful signal at user k while causing less interference at user ¯i(cf. (4.42)).
Denote byγk1,1 the SINR values achieved by settingα12=α21= 1. To show that the corresponding rates belong to the Pareto boundary, we solve the following optimization problem:
maximizeγ1 (4.43)
such that 0≤ζk ≤ ak
ak+bk, k= 1,2 γ2≥γ21,1
98 Chapter 4 Coordinated MISO IC using the VSINR Framework
This can be formalized as the following convex optimization problem:
minimize −t (4.44)
such that 0≤ζk≤ ak
This problem is strictly feasible and consequently Slater’s condition for strong duality holds [55].
Letµi, i= 1, . . . ,3 be the Langrange multipliers associated with the pos-itivity constraints,ξk, k= 1,2 the Lagrange multipliers associated with the upper bounds on theζk, andλk, k= 1,2 the Lagrange multipliers associated with the SINR constraints, the corresponding Karush-Kuhn-Tucker (KKT) conditions [55] are given by:
−1−µ3+λ2(1 +ζ2c2) = 0 values, together with the values of t and the Lagrange multipliers given in equation (4.46) below provide a consistent solution of the KKT conditions.
This guarantees optimality. Noting that the optimal value of problem (4.43)
4.C 99
is indeed that achieved by our algorithm completes the proof.
µ1 =µ2=µ3= 0, ξ1 =ξ2 = 0, λ2= 1
1 +ζ2c2, λ1= 1
1 +ζ2c2
(a1+b1(1 +c1))2(a2+b2(1 +c2)2) (a2+b2(1 +c2))2(a1+b1(1 +c1)2),
t=γ11,1. (4.46)
100 Chapter 4 Coordinated MISO IC using the VSINR Framework
Chapter 5
Cooperative Network MIMO with imperfect CSIT sharing
5.1 Introduction
In this and the following chapter, we move away from the interference chan-nel to a chanchan-nel where joint MIMO precoding across distant transmitters to cooperatively serve the set of mobile users is possible, i.e. user data is shared by several transmitters. Such a scheme, also referred to as so-called network MIMO, was dealt with in [10], [11] for example. This situation is illustrated in Fig. 5.1.
In the downlink scenario, implementation of network MIMO requires both data and CSI to be shared by the transmitters, or to be fed back to some central processor which designs the transmission and informs the base stations of which precoding solutions shall be used. Data and CSI sharing comes, however, at a cost of delay, feedback and backhaul resources. One way to reduce the delays or have a more efficient use of the backhaul is to reduce how much CSI needs to be shared. Thus, assuming that the user data is conveniently routed to all concerned transmitters, but assuming the transmitters obtain local CSIT only, we obtain a MIMO channel with a novel CSIT model where the different transmitters do not have the same vision of the downlink channel. To the best of our knowledge this problem has not yet been investigated, despite its strong relevance in practical situations:
101
102 Chapter 5 Cooperative Network MIMO with distributed CSIT in fact, previous work on multi-transmitter MIMO precoding assumes that either perfect [10], [11] or limited CSIT [56] is available and shared among all transmitters. Instances of such a new distributed CSIT situation are described below.
A first case occurs in the context of a reciprocal system, in which CSIT is acquired from uplink transmissions. Transmitters thus only learn the channels between themselves and the users, but not those between the other transmitters and the users. As mentioned before, full CSI sharing may be too costly, but perhaps some statistical information about the unknown links may be affordable. This is essentially the same level of CSI assumed in Chapter 4 in the context of the MISO interference channel. In fact our initial investigation in the context of distributed CSIT is for this setup and extends the VSINR idea to the joint transmission case. This is presented in Section 5.3 below.
In a second instance of a feedback model, we consider that the different receivers broadcast the CSI estimates (which they have obtained over the downlink) over the air, and each transmitter attempts to decode the said information independently (see the framework proposed in [57]). Thus, de-pending on the distance between a receiver and each base station, a given base station may decode successfully the totality or part of the CSI fed back. In another instance of partially shared CSIT, a station may decode completely the CSI feedback of a subset of users, and forward subquantized versions of it to neighboring bases. Note that both approaches lead to i) a reduction in CSI exchange, and ii) different representations of the same channel at the different transmitters.
Ideally, the different transmitters would have to conciliate their views in order to design a consistent set of precoding vectors that will maximize a performance metric at the user side, despite possible differences in their esti-mated CSIT. This problem can be categorized as a so-called“team-decision problem” or a decentralized statistical decision making problem [19, 58].
More generally, in such problems, i) each decision maker (here, transmitter) has different but correlated information about the underlying uncertainty in the channel state, and ii) the transmitters need to act in a coordinated man-ner in order to realize the common payoff (which could be for example, the average sum rate). Such a scheme offers the possibility for reduction in com-munication requirements, at the expense of performance reduction [59], yet is expected to perform better than a framework where the decision makers simply ignore the differences in their view of the channel state.
The contributions in this chapter are as follows:
5.2 System Model 103
• For the TDD CSI acquisition scheme, we propose a suboptimal solution for the MISO case based on the virtual SINR scheme introduced in the previous chapter. We refer to this approach as alayered virtual SINR scheme, as it relies on viewing the broadcast channel with distributed CSIT as a superposition of interference channels.
• The more general team decision problem corresponding to cooperative network MIMO with imperfect CSIT sharing is presented.
• We propose a new distributed CSIT framework for quantized feedback, which allows a simpler formulation of the above team decision problem.
• We investigate the best transmit strategy (here beamforming) to be adopted by the transmitters in this framework, detailing it for the 2×2 case.
• We study via Monte Carlo simulations the performance gap to a sce-nario with centralized, yet still inaccurate, CSIT.
5.2 System Model
Consider a set of N base stations communicating with K mobile stations.
We assume the transmitters have Nt ≥1 antennas each, whereas receivers have a single antenna each.
We use the channel notation introduced in Chapter 2, so that hk,j ∈ C1×Nt corresponds to the channel between transmitter j and receiver k and hk = [h4 k,1 . . . hk,N] ∈ C1×N Nt, receiver k’s whole channel. hk,j ∈ CN
0, σkj2 INt
.
The signal received at mobile stationkis given by:
yk=hkx+nk, (5.1)
wherex∈CN Nt×1 is the concatenated transmit signal sent by all transmit-ters and nk∼ CN(0, σ2) is the noise at that receiver.
Multi-transmitter cooperative processing in the form of joint linear pre-coding with per-transmitter power constraints is adopted. Thus, x can be expressed as:
x=Ws=
K
X
k=1
wksk, (5.2)
104 Chapter 5 Cooperative Network MIMO with distributed CSIT
...
...
h1
BSN BS1
) 1 (
ˆh1 ˆ(1)
hK
...
hˆ1(N) ˆ(N)hK
...
MS1 MSK
...
hK
Figure 5.1: Cooperative MIMO channel with imperfect CSIT sharing setup, withN base stations,K mobile stations.
wheres∈CK×1is the vector of transmit symbols, its entries are assumed to be independent andCN(0,1). The precoding matrix isW= [w1 . . .wK]∈ CN Nt×K, where wk = [wk,1;. . .;wk,N] is the beamforming vector corre-sponding to user k’s symbol, wk,j ∈ CNt×1 corresponding to Transmitter j’s precoding. Defining Vj as [w1,j . . . wK,j], i.e. as the precoding matrix used at transmitterj,W may be alternatively written as
V1 . . . VN
. (5.3)
The signal transmitted byBSj,xj, is thus
xj =
K
X
k=1
wk,jsk=
K
X
k=1
√pkjuk,jsk, (5.4)
where wk,j = √
pkjuk,j, kuk,jk = 1, and pkj is the power allocated by transmitterj to serving user k.
5.2 System Model 105
Its power constraint is given by:
kVjk2F=
K
X
k=1
kwk,jk2 =
K
X
k=1
pkj ≤P, ∀j= 1, . . . , N. (5.5) Finally, the rate achieved at userk is equal to
rk = log2(1 +γk), (5.6)
where the SINR γk is equal to
γk = |PN
j=1hk,jwk,j|2 σ2+P
¯k=1,...,K,k6=k¯ |PN
j=1hk,jw¯k,j|2. (5.7) As stated in the introduction, the user information symbolssare routed to the multiple cooperating base stations. However, the CSI is not fully shared, and the design of the precoding will need to take this into consider-ation. Details of the distributed CSI knowledge follow.
5.2.1 Distributed CSIT
Previous work on multi-transmitter MIMO precoding assumes that either i) perfect CSIT is shared and available at all transmitters [10, 60], [11] or ii) limited CSIT is available, yet common to all transmitters e.g. [56]. Here we argue that a more general and realistic setup is one where the CSI feedback is designed in such a way that different transmitters end up with different representations of the channel: for instance it is likely that users which are relatively closer to some transmitters will be able to convey more precise information about their channel state to these in an FDD system, whereas in a TDD system in which CSI is acquired from estimating the channel on the uplink, the different transmitters will naturally only be able to learn the channels between themselves and the users but not those from the other transmitters to the users.
The benefit of such distributed CSIT schemes is a reduction in signaling with respect to the scheme where all transmitters must achieve the same state of CSI knowledge, hence potentially a greater scalability of multi-transmitter MIMO cooperation.
The distributed CSI model for quantized channel feedback is shown in Figure 5.2, where transmitterj’s knowledge ofhkis represented by its quan-tized version hˆ(j)k . For the TDD channel estimation case, one can think of hˆ(j)k as being equal to [0, . . . ,hk,j. . . ,0]: hk,j is known perfectly at transmit-ter j but none of the other channel coefficients inhk, for all k (sufficiently close to that transmitter).
106 Chapter 5 Cooperative Network MIMO with distributed CSIT
hk
1(.) Qk
k(.) QN
k(.) Qj
. . .
. . .
) 1
ˆ(
hk
)
ˆ(j
hk
)
ˆ(N
hk
Figure 5.2: Distributed CSI model: each CSI vector is seen through a dif-ferent quantization filter at each base station.